We don't know for certain why she was sent there, since her parents (presumably the ones who had her committed) are dead now. This is introduced toward the end of Twilight, when James and Bella meet at the ballet studio. He explains to her that Alice was another big hunt for him, but the vampire who watched over her changed her into a vampire so that James wouldn't be able to kill her.

Up until that point, Alice's earliest memories were as a vampire; she couldn't remember anything of her human life. We can infer to a certain extent that her talent for seeing the future might have been a reason that she was committed to the asylum in the first place, since we know that the vampires with special talents brought those talents with them from their former lives as humans. Perhaps Alice's parents looked at her talent as a sign of psychosis, and had her committed for it.

During New Moon, Alice searches for clues to her past, but she comes up with little new information.

It is implied that Alice was having visions when she was human. Many of the people in the village or town thought she was crazy and insane so they threw Alice into the mental rehabilitation center. This is also how she was sort of turned into a vampire. I think the one who changed Alice saw her potential as vampire and changed her.

Although it is not known for certain, based on Jame's rant at the end of twilight Alice was able to see the future as a human. Since she was human back in the day, everyone probably thought she was a lunatic and lock her up.

James made a reference to how people in asylums were treated-and it was not pleasant which might be the reason as a vampire she wasn't able to recall anything-she was sedated for most of her human life.